Theorem coe1tmmul2 21357

Description: Coefficient vector of a polynomial multiplied on the right by a term. (Contributed by Stefan O'Rear, 27-Mar-2015.)

Hypotheses

Ref	Expression
coe1tm.z	⊢ 0 = (0g‘𝑅)
coe1tm.k	⊢ 𝐾 = (Base‘𝑅)
coe1tm.p	⊢ 𝑃 = (Poly1‘𝑅)
coe1tm.x	⊢ 𝑋 = (var1‘𝑅)
coe1tm.m	⊢ · = ( ·𝑠 ‘𝑃)
coe1tm.n	⊢ 𝑁 = (mulGrp‘𝑃)
coe1tm.e	⊢ ↑ = (.g‘𝑁)
coe1tmmul.b	⊢ 𝐵 = (Base‘𝑃)
coe1tmmul.t	⊢ ∙ = (.r‘𝑃)
coe1tmmul.u	⊢ × = (.r‘𝑅)
coe1tmmul.a	⊢ (𝜑 → 𝐴 ∈ 𝐵)
coe1tmmul.r	⊢ (𝜑 → 𝑅 ∈ Ring)
coe1tmmul.c	⊢ (𝜑 → 𝐶 ∈ 𝐾)
coe1tmmul.d	⊢ (𝜑 → 𝐷 ∈ ℕ0)

Assertion

Ref	Expression
coe1tmmul2	⊢ (𝜑 → (coe1‘(𝐴 ∙ (𝐶 · (𝐷 ↑ 𝑋)))) = (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 )))

Distinct variable groups:   𝑥, 0   𝑥,𝐶   𝑥,𝐷   𝑥,𝐾   𝑥, ↑   𝑥,𝐴   𝑥,𝑁   𝑥,𝑃   𝑥,𝑋   𝜑,𝑥   𝑥,𝑅   𝑥, ·   𝑥, ×   𝑥, ∙

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem coe1tmmul2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		coe1tmmul.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
2		coe1tmmul.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
3		coe1tmmul.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐾)
4		coe1tmmul.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ ℕ0)
5		coe1tm.k	. . . . 5 ⊢ 𝐾 = (Base‘𝑅)
6		coe1tm.p	. . . . 5 ⊢ 𝑃 = (Poly1‘𝑅)
7		coe1tm.x	. . . . 5 ⊢ 𝑋 = (var1‘𝑅)
8		coe1tm.m	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑃)
9		coe1tm.n	. . . . 5 ⊢ 𝑁 = (mulGrp‘𝑃)
10		coe1tm.e	. . . . 5 ⊢ ↑ = (.g‘𝑁)
11		coe1tmmul.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑃)
12	5, 6, 7, 8, 9, 10, 11	ply1tmcl 21353	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → (𝐶 · (𝐷 ↑ 𝑋)) ∈ 𝐵)
13	1, 3, 4, 12	syl3anc 1369	. . 3 ⊢ (𝜑 → (𝐶 · (𝐷 ↑ 𝑋)) ∈ 𝐵)
14		coe1tmmul.t	. . . 4 ⊢ ∙ = (.r‘𝑃)
15		coe1tmmul.u	. . . 4 ⊢ × = (.r‘𝑅)
16	6, 14, 15, 11	coe1mul 21351	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐴 ∈ 𝐵 ∧ (𝐶 · (𝐷 ↑ 𝑋)) ∈ 𝐵) → (coe1‘(𝐴 ∙ (𝐶 · (𝐷 ↑ 𝑋)))) = (𝑥 ∈ ℕ0 ↦ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)))))))
17	1, 2, 13, 16	syl3anc 1369	. 2 ⊢ (𝜑 → (coe1‘(𝐴 ∙ (𝐶 · (𝐷 ↑ 𝑋)))) = (𝑥 ∈ ℕ0 ↦ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)))))))
18		eqeq2 2750	. . . 4 ⊢ ((((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶) = if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 ) → ((𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶) ↔ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 )))
19		eqeq2 2750	. . . 4 ⊢ ( 0 = if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 ) → ((𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = 0 ↔ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 )))
20		coe1tm.z	. . . . . . 7 ⊢ 0 = (0g‘𝑅)
21	1	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝑅 ∈ Ring)
22		ringmnd 19708	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
23	21, 22	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝑅 ∈ Mnd)
24		ovex 7288	. . . . . . . 8 ⊢ (0...𝑥) ∈ V
25	24	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (0...𝑥) ∈ V)
26		simprr 769	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝐷 ≤ 𝑥)
27	4	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝐷 ∈ ℕ0)
28		simprl 767	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝑥 ∈ ℕ0)
29		nn0sub 12213	. . . . . . . . . 10 ⊢ ((𝐷 ∈ ℕ0 ∧ 𝑥 ∈ ℕ0) → (𝐷 ≤ 𝑥 ↔ (𝑥 − 𝐷) ∈ ℕ0))
30	27, 28, 29	syl2anc 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (𝐷 ≤ 𝑥 ↔ (𝑥 − 𝐷) ∈ ℕ0))
31	26, 30	mpbid 231	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (𝑥 − 𝐷) ∈ ℕ0)
32	27	nn0ge0d 12226	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 0 ≤ 𝐷)
33		nn0re 12172	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ0 → 𝑥 ∈ ℝ)
34	33	ad2antrl 724	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝑥 ∈ ℝ)
35	4	nn0red 12224	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ ℝ)
36	35	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝐷 ∈ ℝ)
37	34, 36	subge02d 11497	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (0 ≤ 𝐷 ↔ (𝑥 − 𝐷) ≤ 𝑥))
38	32, 37	mpbid 231	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (𝑥 − 𝐷) ≤ 𝑥)
39		fznn0 13277	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0 → ((𝑥 − 𝐷) ∈ (0...𝑥) ↔ ((𝑥 − 𝐷) ∈ ℕ0 ∧ (𝑥 − 𝐷) ≤ 𝑥)))
40	39	ad2antrl 724	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → ((𝑥 − 𝐷) ∈ (0...𝑥) ↔ ((𝑥 − 𝐷) ∈ ℕ0 ∧ (𝑥 − 𝐷) ≤ 𝑥)))
41	31, 38, 40	mpbir2and 709	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (𝑥 − 𝐷) ∈ (0...𝑥))
42	1	ad2antrr 722	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑅 ∈ Ring)
43		eqid 2738	. . . . . . . . . . . . 13 ⊢ (coe1‘𝐴) = (coe1‘𝐴)
44	43, 11, 6, 5	coe1f 21292	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝐵 → (coe1‘𝐴):ℕ0⟶𝐾)
45	2, 44	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (coe1‘𝐴):ℕ0⟶𝐾)
46	45	ad2antrr 722	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (coe1‘𝐴):ℕ0⟶𝐾)
47		elfznn0 13278	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0...𝑥) → 𝑦 ∈ ℕ0)
48	47	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑦 ∈ ℕ0)
49	46, 48	ffvelrnd 6944	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → ((coe1‘𝐴)‘𝑦) ∈ 𝐾)
50		eqid 2738	. . . . . . . . . . . . 13 ⊢ (coe1‘(𝐶 · (𝐷 ↑ 𝑋))) = (coe1‘(𝐶 · (𝐷 ↑ 𝑋)))
51	50, 11, 6, 5	coe1f 21292	. . . . . . . . . . . 12 ⊢ ((𝐶 · (𝐷 ↑ 𝑋)) ∈ 𝐵 → (coe1‘(𝐶 · (𝐷 ↑ 𝑋))):ℕ0⟶𝐾)
52	13, 51	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (coe1‘(𝐶 · (𝐷 ↑ 𝑋))):ℕ0⟶𝐾)
53	52	ad2antrr 722	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (coe1‘(𝐶 · (𝐷 ↑ 𝑋))):ℕ0⟶𝐾)
54		fznn0sub 13217	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0...𝑥) → (𝑥 − 𝑦) ∈ ℕ0)
55	54	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (𝑥 − 𝑦) ∈ ℕ0)
56	53, 55	ffvelrnd 6944	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)) ∈ 𝐾)
57	5, 15	ringcl 19715	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘𝐴)‘𝑦) ∈ 𝐾 ∧ ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)) ∈ 𝐾) → (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))) ∈ 𝐾)
58	42, 49, 56, 57	syl3anc 1369	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))) ∈ 𝐾)
59	58	fmpttd 6971	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)))):(0...𝑥)⟶𝐾)
60	1	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)})) → 𝑅 ∈ Ring)
61	3	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)})) → 𝐶 ∈ 𝐾)
62	4	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)})) → 𝐷 ∈ ℕ0)
63		eldifi 4057	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)}) → 𝑦 ∈ (0...𝑥))
64	63, 54	syl 17	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)}) → (𝑥 − 𝑦) ∈ ℕ0)
65	64	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)})) → (𝑥 − 𝑦) ∈ ℕ0)
66		eldifsn 4717	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)}) ↔ (𝑦 ∈ (0...𝑥) ∧ 𝑦 ≠ (𝑥 − 𝐷)))
67		simplrl 773	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑥 ∈ ℕ0)
68	67	nn0cnd 12225	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑥 ∈ ℂ)
69	47	nn0cnd 12225	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ (0...𝑥) → 𝑦 ∈ ℂ)
70	69	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑦 ∈ ℂ)
71	68, 70	nncand 11267	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (𝑥 − (𝑥 − 𝑦)) = 𝑦)
72	71	eqcomd 2744	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → 𝑦 = (𝑥 − (𝑥 − 𝑦)))
73		oveq2 7263	. . . . . . . . . . . . . . . 16 ⊢ (𝐷 = (𝑥 − 𝑦) → (𝑥 − 𝐷) = (𝑥 − (𝑥 − 𝑦)))
74	73	eqeq2d 2749	. . . . . . . . . . . . . . 15 ⊢ (𝐷 = (𝑥 − 𝑦) → (𝑦 = (𝑥 − 𝐷) ↔ 𝑦 = (𝑥 − (𝑥 − 𝑦))))
75	72, 74	syl5ibrcom 246	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (𝐷 = (𝑥 − 𝑦) → 𝑦 = (𝑥 − 𝐷)))
76	75	necon3d 2963	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (𝑦 ≠ (𝑥 − 𝐷) → 𝐷 ≠ (𝑥 − 𝑦)))
77	76	impr 454	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ (𝑦 ∈ (0...𝑥) ∧ 𝑦 ≠ (𝑥 − 𝐷))) → 𝐷 ≠ (𝑥 − 𝑦))
78	66, 77	sylan2b 593	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)})) → 𝐷 ≠ (𝑥 − 𝑦))
79	20, 5, 6, 7, 8, 9, 10, 60, 61, 62, 65, 78	coe1tmfv2 21356	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)})) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)) = 0 )
80	79	oveq2d 7271	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)})) → (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))) = (((coe1‘𝐴)‘𝑦) × 0 ))
81	5, 15, 20	ringrz 19742	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘𝐴)‘𝑦) ∈ 𝐾) → (((coe1‘𝐴)‘𝑦) × 0 ) = 0 )
82	42, 49, 81	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ (0...𝑥)) → (((coe1‘𝐴)‘𝑦) × 0 ) = 0 )
83	63, 82	sylan2 592	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)})) → (((coe1‘𝐴)‘𝑦) × 0 ) = 0 )
84	80, 83	eqtrd 2778	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) ∧ 𝑦 ∈ ((0...𝑥) ∖ {(𝑥 − 𝐷)})) → (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))) = 0 )
85	84, 25	suppss2 7987	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → ((𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)))) supp 0 ) ⊆ {(𝑥 − 𝐷)})
86	5, 20, 23, 25, 41, 59, 85	gsumpt 19478	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = ((𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))‘(𝑥 − 𝐷)))
87		fveq2 6756	. . . . . . . . 9 ⊢ (𝑦 = (𝑥 − 𝐷) → ((coe1‘𝐴)‘𝑦) = ((coe1‘𝐴)‘(𝑥 − 𝐷)))
88		oveq2 7263	. . . . . . . . . 10 ⊢ (𝑦 = (𝑥 − 𝐷) → (𝑥 − 𝑦) = (𝑥 − (𝑥 − 𝐷)))
89	88	fveq2d 6760	. . . . . . . . 9 ⊢ (𝑦 = (𝑥 − 𝐷) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)) = ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − (𝑥 − 𝐷))))
90	87, 89	oveq12d 7273	. . . . . . . 8 ⊢ (𝑦 = (𝑥 − 𝐷) → (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))) = (((coe1‘𝐴)‘(𝑥 − 𝐷)) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − (𝑥 − 𝐷)))))
91		eqid 2738	. . . . . . . 8 ⊢ (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)))) = (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))
92		ovex 7288	. . . . . . . 8 ⊢ (((coe1‘𝐴)‘(𝑥 − 𝐷)) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − (𝑥 − 𝐷)))) ∈ V
93	90, 91, 92	fvmpt 6857	. . . . . . 7 ⊢ ((𝑥 − 𝐷) ∈ (0...𝑥) → ((𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))‘(𝑥 − 𝐷)) = (((coe1‘𝐴)‘(𝑥 − 𝐷)) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − (𝑥 − 𝐷)))))
94	41, 93	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → ((𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))‘(𝑥 − 𝐷)) = (((coe1‘𝐴)‘(𝑥 − 𝐷)) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − (𝑥 − 𝐷)))))
95	28	nn0cnd 12225	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝑥 ∈ ℂ)
96	27	nn0cnd 12225	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝐷 ∈ ℂ)
97	95, 96	nncand 11267	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (𝑥 − (𝑥 − 𝐷)) = 𝐷)
98	97	fveq2d 6760	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − (𝑥 − 𝐷))) = ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝐷))
99	3	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → 𝐶 ∈ 𝐾)
100	20, 5, 6, 7, 8, 9, 10	coe1tmfv1 21355	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐶 ∈ 𝐾 ∧ 𝐷 ∈ ℕ0) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝐷) = 𝐶)
101	21, 99, 27, 100	syl3anc 1369	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘𝐷) = 𝐶)
102	98, 101	eqtrd 2778	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − (𝑥 − 𝐷))) = 𝐶)
103	102	oveq2d 7271	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (((coe1‘𝐴)‘(𝑥 − 𝐷)) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − (𝑥 − 𝐷)))) = (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶))
104	86, 94, 103	3eqtrd 2782	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ ℕ0 ∧ 𝐷 ≤ 𝑥)) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶))
105	104	anassrs 467	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ 𝐷 ≤ 𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶))
106	1	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 𝑅 ∈ Ring)
107	3	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 𝐶 ∈ 𝐾)
108	4	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 𝐷 ∈ ℕ0)
109	54	ad2antll 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (𝑥 − 𝑦) ∈ ℕ0)
110	54	nn0red 12224	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (0...𝑥) → (𝑥 − 𝑦) ∈ ℝ)
111	110	ad2antll 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (𝑥 − 𝑦) ∈ ℝ)
112	33	ad2antlr 723	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 𝑥 ∈ ℝ)
113	35	ad2antrr 722	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 𝐷 ∈ ℝ)
114	47	ad2antll 725	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 𝑦 ∈ ℕ0)
115	114	nn0ge0d 12226	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 0 ≤ 𝑦)
116	47	nn0red 12224	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (0...𝑥) → 𝑦 ∈ ℝ)
117	116	ad2antll 725	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 𝑦 ∈ ℝ)
118	112, 117	subge02d 11497	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (0 ≤ 𝑦 ↔ (𝑥 − 𝑦) ≤ 𝑥))
119	115, 118	mpbid 231	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (𝑥 − 𝑦) ≤ 𝑥)
120		simprl 767	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → ¬ 𝐷 ≤ 𝑥)
121	112, 113	ltnled 11052	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (𝑥 < 𝐷 ↔ ¬ 𝐷 ≤ 𝑥))
122	120, 121	mpbird 256	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 𝑥 < 𝐷)
123	111, 112, 113, 119, 122	lelttrd 11063	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (𝑥 − 𝑦) < 𝐷)
124	111, 123	gtned 11040	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → 𝐷 ≠ (𝑥 − 𝑦))
125	20, 5, 6, 7, 8, 9, 10, 106, 107, 108, 109, 124	coe1tmfv2 21356	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)) = 0 )
126	125	oveq2d 7271	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))) = (((coe1‘𝐴)‘𝑦) × 0 ))
127	45	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (coe1‘𝐴):ℕ0⟶𝐾)
128	127, 114	ffvelrnd 6944	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → ((coe1‘𝐴)‘𝑦) ∈ 𝐾)
129	106, 128, 81	syl2anc 583	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (((coe1‘𝐴)‘𝑦) × 0 ) = 0 )
130	126, 129	eqtrd 2778	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ (¬ 𝐷 ≤ 𝑥 ∧ 𝑦 ∈ (0...𝑥))) → (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))) = 0 )
131	130	anassrs 467	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ¬ 𝐷 ≤ 𝑥) ∧ 𝑦 ∈ (0...𝑥)) → (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))) = 0 )
132	131	mpteq2dva 5170	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ¬ 𝐷 ≤ 𝑥) → (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)))) = (𝑦 ∈ (0...𝑥) ↦ 0 ))
133	132	oveq2d 7271	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ¬ 𝐷 ≤ 𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )))
134	1, 22	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Mnd)
135	20	gsumz 18389	. . . . . . 7 ⊢ ((𝑅 ∈ Mnd ∧ (0...𝑥) ∈ V) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )) = 0 )
136	134, 24, 135	sylancl 585	. . . . . 6 ⊢ (𝜑 → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )) = 0 )
137	136	ad2antrr 722	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ¬ 𝐷 ≤ 𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ 0 )) = 0 )
138	133, 137	eqtrd 2778	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℕ0) ∧ ¬ 𝐷 ≤ 𝑥) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = 0 )
139	18, 19, 105, 138	ifbothda 4494	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ0) → (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦))))) = if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 ))
140	139	mpteq2dva 5170	. 2 ⊢ (𝜑 → (𝑥 ∈ ℕ0 ↦ (𝑅 Σg (𝑦 ∈ (0...𝑥) ↦ (((coe1‘𝐴)‘𝑦) × ((coe1‘(𝐶 · (𝐷 ↑ 𝑋)))‘(𝑥 − 𝑦)))))) = (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 )))
141	17, 140	eqtrd 2778	1 ⊢ (𝜑 → (coe1‘(𝐴 ∙ (𝐶 · (𝐷 ↑ 𝑋)))) = (𝑥 ∈ ℕ0 ↦ if(𝐷 ≤ 𝑥, (((coe1‘𝐴)‘(𝑥 − 𝐷)) × 𝐶), 0 )))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  Vcvv 3422   ∖ cdif 3880  ifcif 4456  {csn 4558   class class class wbr 5070   ↦ cmpt 5153  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255  ℂcc 10800  ℝcr 10801  0cc0 10802   < clt 10940   ≤ cle 10941   − cmin 11135  ℕ₀cn0 12163  ...cfz 13168  Basecbs 16840  ._rcmulr 16889   ·_𝑠 cvsca 16892  0_gc0g 17067   Σ_g cgsu 17068  Mndcmnd 18300  ._gcmg 18615  mulGrpcmgp 19635  Ringcrg 19698  var₁cv1 21257  Poly₁cpl1 21258  coe₁cco1 21259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-ofr 7512  df-om 7688  df-1st 7804  df-2nd 7805  df-supp 7949  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-pm 8576  df-ixp 8644  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-fsupp 9059  df-oi 9199  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-z 12250  df-dec 12367  df-uz 12512  df-fz 13169  df-fzo 13312  df-seq 13650  df-hash 13973  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-mulr 16902  df-sca 16904  df-vsca 16905  df-tset 16907  df-ple 16908  df-0g 17069  df-gsum 17070  df-mre 17212  df-mrc 17213  df-acs 17215  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-mhm 18345  df-submnd 18346  df-grp 18495  df-minusg 18496  df-sbg 18497  df-mulg 18616  df-subg 18667  df-ghm 18747  df-cntz 18838  df-cmn 19303  df-abl 19304  df-mgp 19636  df-ur 19653  df-ring 19700  df-subrg 19937  df-lmod 20040  df-lss 20109  df-psr 21022  df-mvr 21023  df-mpl 21024  df-opsr 21026  df-psr1 21261  df-vr1 21262  df-ply1 21263  df-coe1 21264

This theorem is referenced by:  coe1tmmul2fv  21359  coe1sclmul2  21365